A note on the algorithm of symplectic integrators

For a Hamiltonian that can be separated into N+1(N\geq 2) integrable parts, four algorithms can be built for a symplectic integrator. This research compares these algorithms for the first and second order integrators. We found that they have similar local truncation errors represented by error Hamiltonian but rather different numerical stability. When the computation of the main part of the Hamiltonian, H ₀, is not expensive, we recommend to use S ^* type algorithm, which cuts the calculation of the H ₀ system into several small time steps as Malhotra(1991) did. As to the order of the N+1 parts in one step calculation, we found that from the large to small would get a slower error accumulation. This revised version was published online in July 2006 with corrections to the Cover Date.     

Tidal deformations of the earth from VLBI observations

Based on an analysis of the VLBI observations performed in 1985?C2010 within the framework of international geodetic programs on global networks of stations, we have obtained statistically significant corrections to the parameters of lunisolar tides??the nominal complex Love/Shida numbers. The new integral (frequency-independent) values of these parameters (in 10^?4) are h ⁽⁰⁾ = (6113 ± 3) ? (33 ± 2)i, l ⁽⁰⁾ = (843 ± 1) ? (5 ± 2)i for a total tide; h ⁽⁰⁾ = (6106 ± 3) ? (10 ± 6)i, l ⁽⁰⁾ = (843 ± 1) ? (8 ± 1)i for diurnal tides; and h ⁽⁰⁾ = (6106 ± 3) ? (24 ± 3)i, l ⁽⁰⁾ = (843 ± 1) + (3 ± 1)i for semi-diurnal tides. We have detected a new effect of asymmetry in the horizontal tidal displacements in the direction of tectonic motions for 50 VLBI stations. We have determined upper limits for the influence of the frequency-dependent resonance effects whose estimation accuracy is limited by an abundance of close frequencies in their harmonic expansion. The influence of the transfer function for tides on the VLBI observations has turned out to be lower than the measurement accuracy. In future, positional GPS/GLONASS measurements are planned to be used to refine the resonance effects and the transfer function.     

Comparing maps to symplectic integrators in a galactic type Hamiltonian

We obtain thex - p _xPoincare phase plane for a two dimensional, resonant, galactic type Hamiltonian using conventional numerical integration, a second order symplectic integrator and a map based on the averaged Hamiltonian. It is found that all three methods give good results, for small values of the perturbation parameter, while the symplectic integrator does a better job than the mapping, for large perturbations. The dynamical spectra are used to distinguish between regular and chaotic motion.     

Third and fourth order resonances in Hamiltonian systems

The stability of the origin of an autonomous Hamiltonian system is investigated when the system possesses a third or fourth-order resonance.H ₂, the quadratic part ofH isH ₂=ⁿ _{i=1 i} J _i and the resonance condition is ⁿ _i=1 k _{i i} where thek 0,i = 1, 2, ...,n are the natural or fundamental frequencies. It is shown that the only case in which the origin can be unstable is ifk _i0,i=1,2,...,n. The condition for instability is then given in terms of the coefficients of the higher order terms in the Hamiltonian. The transfer of energy between modes is also investigated when a near-resonant condition exists.     

Do Average Hamiltonians Exist?

The word "average" and its variations became popular in the sixties and implicitly carried the idea that "averaging" methods lead to "average" Hamiltonians. However, given the Hamiltonian H = H₀(J) + ∈R(θ, J), (∈ < < 1), the problem of transforming it into a new Hamiltonian H^* (J^*) (dependent only on the new actions J^*), through a canonical transformation given by zero-average trigonometrical series has no general solution at orders higher than the first. This revised version was published online in July 2006 with corrections to the Cover Date.     

Effect of oblateness, perturbations, radiation and varying masses on the stability of equilibrium points in the restricted three-body problem

This paper investigates the combined effect of small perturbations ε,ε′ in the Coriolis and centrifugal forces, radiation pressure q _i , and changing oblateness of the primaries A _i (t) (i=1,2) on the stability of equilibrium points in the restricted three body problem in which the primaries is a supergiant eclipsing binary system which consists of a pair of bright oblate stars having the appearance of a giant peanut in space and their masses assumed to vary with time in the absence of reactive forces. The equations of motion are derived and the equilibrium points are obtained. For the autonomized system, it is seen that there are more than a pair of the triangular points as κ→∞; κ being the arbitrary sum of the masses of the primaries. In the case of the collinear points, two additional equilibrium points exist on the line joining the primaries when simultaneously κ+ε′<0 and both primaries are oblate, i.e., 0<α _i ?1. So there are five collinear equilibrium points in this case. Two non-planar equilibrium points exist for κ>1. Hence, there are at least nine equilibrium points of the system. The stability of these points is explored analytically and numerically. It is seen that the collinear and triangular points are stable with respect to certain conditions controlled by κ while the non-planar equilibrium points are unstable.     

On The Ideal Resonance Problem

The Ideal Resonance Problem in its normal form is defined by the Hamiltonian (1) $$F = A (y) + 2B (y) sin^2 x$$ with (2) $$A = 0(1),B = 0(\varepsilon )$$ where ? is a small parameter, andx andy a pair of canonically conjugate variables. A solution to 0(?^1/2) has been obtained by Garfinkel (1966) and Jupp (1969). An extension of the solution to 0(?) is now in progress in two papers ([Garfinkel and Williams] and [Hori and Garfinkel]), using the von Zeipel and the Hori-Lie perturbation methods, respectively. In the latter method, the unperturbed motion is that of a simple pendulum. The character of the motion depends on the value of theresonance parameter α, defined by (3) $$\alpha = - A\prime /|4A\prime \prime B\prime |^{1/2} $$ forx=0. We are concerned here withdeep resonance, (4) $$\alpha< \varepsilon ^{ - 1/4} ,$$ where the classical solution with a critical divisor is not admissible. The solution of the perturbed problem would provide a theoretical framework for an attack on a problem of resonance in celestial mechanics, if the latter is reducible to the Ideal form: The process of reduction involves the following steps: (1) the ration ₁/n₂ of the natural frequencies of the motion generates a sequence. (5) $$n_1 /n_2 \sim \left\{ {Pi/qi} \right\},i = 1, 2 ...$$ of theconvergents of the correspondingcontinued fraction, (2) for a giveni, the class ofresonant terms is defined, and all non-resonant periodic terms are eliminated from the Hamiltonian by a canonical transformation, (3) thedominant resonant term and itscritical argument are calculated, (4) the number of degrees of freedom is reduced by unity by means of a canonical transformation that converts the critical argument into an angular variable of the new Hamiltonian, (5) the resonance parameter α (i) corresponding to the dominant term is then calculated, (6) a search for deep resonant terms is carried out by testing the condition (4) for the function α(i), (7) if there is only one deep resonant term, and if it strongly dominates the remaining periodic terms of the Hamiltonian, the problem is reducible to the Ideal form.     

Molecular and low excitation emission in high mass post-main-sequence nebulae

We have undertaken mapping and spectroscopy of a broad range of type I post-Main-Sequence nebulae in COJ=1→0,J=2→1, andJ=3→2, using the 12 m antenna at Kitt Peak, and the 45 m facility of the Nobeyama Radio Observatory. As a consequence, we find COJ=2→1 emission associated with NGC 3132 and NGC 6445, determine the location of COJ=1→0 emission in the nucleus of NGC 6302, and obtain (for the first time) COJ=3→2 spectroscopy for a substantial cross-section of type I sources. LVG analysis of the results suggests densitiesn(H₂) ～ 10⁴ cm^?3, and velocity gradients dv/dr ～ 2×10² in both NGC 7027 and CRL 618, commensurate with uniform expansion of a constant velocity outflow, whilst for the case of NGC 2346 these values probably exceedn(H₂) ～ 4.0×10⁵ cm^?3. dv/dr ～ 2.6×10³ km s^?1 andT _k ～10² K, implying appreciable compression (and shock heating?) of the CO excitation zone. Hi masses extend over a typical range 0.01<M(Hi)/M _⊙<1, whilst corresponding estimates of the progenitor mass imply 0.7<M _prog/M _⊙<2.3; values significantly in excess of those pertinent for normal PN, although somewhat at the lower end of the type I mass range. COJ=3→2 profiles for CRL 2688 confirm the presence of an extended plateau with width Δv～85 km s^?1, whilst modestJ=3→2 enhancement is also observed for the high-velocity components in NGC 7027. TheJ=3→2 spectrum for NGC 2346 appears to mimic lower-frequency results reasonably closely, confirming the presence of a double-peaked structure towards the core, and predominantly unitary profiles to the north and south, whilst there is also evidence to suggest appreciableJ=3→2 asymmetry in CRL 618 compared to lower-frequency measures. The status of an extended cloud near HB 5 remains uncertain, although this clearly represents a remarkably complex region with velocity span ΔV～50 km s^?1. Our presentJ=3→2 results appear to track lower frequency measures extremely closely, implying local densitiesn(H₂)>3×10³ cm^?3—although temperatures close to theV _lsr of HB 5 are relatively weak, and of orderT _MB (J=3→2)≤0.9 K. Finally, as a result of both this, and previous investigations we find that of type I sources so far observed in CO, some ～42% appear to possess detectable levels of emissionT _r ^* >0.1 K. Similarly, in cross-correlating this data with other results, we note a closely linear relation betweenJ=2→1 antenna temperaturesT _MB, and the surface brightness of H₂ S(1) quadrupole emissionS(H₂)—a trend which appears also to be reflected betweenS(H₂) and corresponding parameters for [Oi], [Oii], [Ni], [Nii], and [Sii]. Such relations almost certainly arise from comparable secular variations in line intensities, although the CO, H₂, and optical emission components are likely to derive from disparate line excitation zones. As a consequence, it is clear that whilst H₂ S(1) emission is probably enhanced as a result of local shock activity, the evidence for post-shock excitation of the CO and optical forbidden lines is at best marginal. Similarly, although it seems likely that CO emission derives from circum-nebular Hi shells with kinetic temperatureT _k ～ 30 K or greater, the predominant fraction of low-excitation emission arises from a mix of charge exchange reactions, nebular stratification and, probably most importantly, the influence of UV shadow zones and associated neutral inclusions.     

Normal form,Lie-Poisson structure and reduction for the Henon-Heiles system

The reduced Henon-Heiles system is investigated as a Hamiltonian dynamical system obtained by applying the normalization of the HamiltonianH=1/2(p ₁ ² +p ₂ ² +q ₁ ² +q ₂ ² )+1/3q ₁ ³ –q ₁ q ₂ ² to fourth-degree terms. The related equations of motion are bi-Hamiltonian and possess the Lie-Poisson structure. Each Lie-Poisson structure possesses an associated Casimir function. When reduced to level sets of these functions, the equations of motion take various symplectic forms. The various reductions give different coordinate representations of the solutions. These coordinate representations are used to seek the simplest representation of the solutions.     

Recent progress in the theory and application of symplectic integrators

In this paper various aspect of symplectic integrators are reviewed. Symplectic integrators are numerical integration methods for Hamiltonian systems which are designed to conserve the symplectic structure exactly as the original flow. There are explicit symplectic schemes for systems of the formH=T(p)+V(q), and implicit schemes for general Hamiltonian systems. As a general property, symplectic integrators conserve the energy quite well and therefore an artificial damping (excitation) caused by the accumulation of the local truncation error cannot occur. Symplectic integrators have been applied to the Kepler problem, the motion of minor bodies in the solar system and the long-term evolution of outer planets.     

TheH-functions of radiative transfer in an active amplifying medium

We obtain two explicit closed form representations of Chandrasekhar'sH-functionsH(z) characterising transfer of radiation in an active amplifying medium corresponding to the dispersion function $$T(z) = 1 - 2z^2 \int\limits_0^1 {Y(x)dx/(z^2 - x^2 ), Y(x)< 0 on [0, 1]} .$$ Their basic properties are derived and the values of theH-functionH(z, ω) whenY(x)=ω/2, are approximately computed for values of ω in the range (?10^?12)–(?10²) and for values ofzε[0, 1].     

Symplectic integrator for general near-integrable Hamiltonian system

In this paper, following the idea of constructing the mixed symplectic integrator (MSI) for a separable Hamiltonian system, we give a low order mixed symplectic integrator for an inseparable, but nearly integrable, Hamiltonian system, Although the difference schemes of the integrators are implicit, they not only have a small truncation error but, due to near integrability, also a faster convergence rate of iterative solution than ordinary implicit integrators, Moreover, these second order integrators are time-reversible.     

Possible H2staggered+ ultraviolet spectrum of Jupiter

An ultraviolet spectral probe for a hydrogen-rich planetary atmosphere, such as that of Jupiter, is suggested, utilizing discrete lines in the H₂^staggered+ 2pπ_u?1sσ_g electronic transition. For the Jovian atmosphere, the dominant mechanism for exciting H₂⁺ to its 2pπ_u state appears to be photoexcitation, principally through absorption of the solar Lyman-α line. We estimate that the Jovian column emission rate of the H₂⁺ 2pπ_u(ν′ = 2, J′ = 1) →1sσ_g(ν″ = 18,J″ = 0) fluorescent line at 1236.6 Å is if1 photon cmsu-2 secsu-1; i.e., that if1 photon secsu-1 of this radiation would strike a 15-cm diameter mirror in a Jupiter fly-by at an impact parameter of 3 × 10⁵km. The critical role of corrections to the Born-Oppenheimer approximation in the use of an H₂⁺ probe is discussed.     

Spectral and photometric studies of polar CRTS CSS 130604 J 215427+155714

We present the results of spectroscopic and photometric studies of a new polar CRTS CSS130604 J 215427+155714, conducted at the telescopes of the SAO RAS. Analysis of the photometric series of observations allowed to clarify the orbital period of the system, P _o = 0_. ^d 0672879 (±0.0000003). We build radial velocity curves and trace the intensity variations in the Hβ and H_γ hydrogen lines and He II λ 4686 ?A ionized heliumline. Based on the Hβ and He II lines we build Doppler maps. It is shown that the line formation region is localized near the Lagrange point. The following parameter estimates of the system are obtained:M ₁ = 0.83 ± 0.10M _⊙, M ₂ = 0.15 ± 0.01M _⊙, q = M ₂/M ₁ = 0.18 ± 0.03, i = 53? ± 5?. Based on the results of spectral, photometric and previously published polarimetric observations the possible geometric model of the system is discussed.     

About Bianchi I with VSL

In this paper we study how to attack, through different techniques, a perfect fluid Bianchi I model with variable G,c and Λ, “but” taking into account the effects of a “c-variable” into the curvature tensor. We study the model under the assumption, div(T)=0. These tactics are: Lie groups method (LM), imposing a particular symmetry, self-similarity (SS), matter collineations (MC) and kinematical self-similarity (KSS). We compare both tactics since they are quite similar (symmetry principles). We arrive to the conclusion that the LM is too restrictive and brings us to get only the flat FRW solution. The SS, MC and KSS approaches bring us to obtain all the quantities depending on (∫ c(t)dt). Therefore, in order to study their behavior we impose some physical restrictions like for example the condition q<0 (accelerating universe). In this way we find that c is a growing time function and Λ is a decreasing time function whose sing depends on the equation of state ω, while the exponents of the scale factor must satisfy the conditions ∑ _i=1 ³ α _i =1 and ∑ _i=1 ³ α _i ² <1, ? ω, i.e. for all equation of state, relaxing in this way the Kasner conditions. The behavior of G depends on two parameters, the equation of state ω and ε, a parameter that controls the behavior of c(t), therefore G may be growing or decreasing. We also show that through the Lie method, there is no difference between to study the field equations under the assumption of a c-var affecting to the curvature tensor which the other one where it is not considered such effects. Nevertheless, it is essential to consider such effects in the cases studied under the SS, MC, and KSS hypotheses.     

A search for H2O and Ch4 in Comet Kohoutek

A spectroscopic search for H₂O and CH₄ in Comet Kohoutek (1973f) was made using a Pepsios interferometer. No evidence was found for either molecule, allowing us to set an upper limit on their production rates (on about 21 January 1974) of Q(H₂O) < 6.2 × 10²⁸ sec^?1 and Q(CH₄) < 2.0 × 10³⁰ sec^?1. If the cometary surface is water-ice, this production rate leads to a product (1 ? A)·(πR₀²) < 2.2 km², where A is the Bond albedo, R₀ is the nuclear radius, and we assume that all the absorbed solar energy is used to evaporate H₂O.     

On the photodissociation of water vapour in the mesosphere

The photodissociation of water vapour in the mesosphere depends on the absorption of solar radiation in the region (175–200 nm) of the O₂ Schumann-Runge band system and also at H-Lyman alpha. The photodissociation products are OH + H, OH + H, O + 2H and H₂ + O at Lyman alpha; the percentages for these four channels are 70, 8, 12 and 10%, respectively, but OH + H is the only channel between 175 and 200 nm. Such proportions lead to a production of H atoms corresponding to practically the total photodissociation of H₂O, while the production of H₂ molecules is only 10% of the H₂O photodissociation by Lyman alpha.The photodissociation frequency (s^?1) at Lyman alpha can be expressed by a simple formula

$\text{J}{}_{\mathrm{<mtext>Ly</mtext><mtext>\alpha </mtext>}}\text{H}{}_2\text{O=4.5 ×10}{}^{\mathrm{?6}}\text{1+0.2}\text{F}{}_{10.7}\text{?65}\text{100}\text{exp}\text{[?4.4 ×10}{}^{\mathrm{?19}}\text{N}{}^{0.917}\text{]}$

where F_{10.7 cm} is the solar radioflux at 10.7 cm and N the total number of O₂ molecules (cm^?2), and when the following conventional value is accepted for the Lyman alpha solar irradiance at the top of the Earth's atmosphere ($\text{Δλ = 3.5}\text{A}\text{?}$) q_Lyα,∞ = 3 × 10¹¹ photons cm^?2 s^1?.The photodissociation frequency for the Schumann-Runge band region is also given for mesospheric conditions by a simple formula

$\text{J}{}_{\mathrm{<mtext>SRB</mtext>}}\text{(}\text{H}{}_2\text{O}\text{) = J}{}_{\mathrm{<mtext>SRB</mtext><mtext>,\infty </mtext>}}\text{(}\text{H}{}_2\text{O}\text{) exp [?10}{}^{\mathrm{?7}}\text{N}{}^{0.35}\text{]}$

where J_SRB,∞(H₂O) = 1.2 × 10^?6 and 1.4 × 10^?6 s^?1 for quiet and active sun conditions, respectively.The precision of both formulae is good, with an uncertainty less than 10%, but their accuracy depends on the accuracy of observational and experimental parameters such as the absolute solar irradiances, the variable transmittance of O₂ and the H₂O effective absorption cross sections. The various uncertainties are discussed. As an example, the absolute values deduced from the above formulae could be decreased by about 25-20% if the possible minimum values of the solar irradiances were used.     

Temperature and solar zenith angle control of D-region positive ion chemistry

A numerical model is utilized to investigate the temperature (T) and solar zenith angle (χ) control of D-region positive ion chemistry between 75 and 90 km. It is assumed that NO^? is the precursor ion in a chain which involves three-body formation of the intermediary cluster ions NO⁺(H₂O)_m?1(X) (m = 1–3), where X can be N₂,O₂, H₂O, or CO₂, switching reactions which convert these weakly bound clusters to hydrates of NO⁺ and reaction of the third hydrate of NO⁺ with H₂O to initiate the chain to form H⁺(H₂O)_n (n = 1–7). Zonal mean and tidal temperatures from rocket observations and theory are synthesized to obtain the best available estimate of mean latitudinal, seasonal and local time variations of temperature in this height region. Relative compositions of NO⁺(H₂O)_m and H⁺(H₂O)_n are found to vary widely over the complete range of realistic conditions; however, the relative ion populations are entirely explicable in terms of the effects of χ and T on the relative life-times of the intermediary ion clusters with respect to recombination, switching and thermal decomposition. For instance, as χ increases (and electron production decreases) beyond 60° for a given temperature, the recombination times of the intermediate ion cluster species lengthen with respect to the formation time of the H⁺ water clusters, causing the relative H⁺ water cluster population to increase and thus raise the level where the cluster ion and NO⁺ concentrations are equal from about 85 km (normal midday) to 90 km. For a given χ the concentrations of NO⁺H₂O and H⁺(H₂O)₄ increase (decrease) for temperatures less than (greater than) 190 and 205 K, respectively. The transition occurs when the temperature becomes sufficiently high that the lifetimes of intermediary ion clusters with respect to thermal decomposition become less than their lifetimes with respect to H₂O switching (which ultimately leads to the third hydrate of NO⁺ and entry into the water chain). At this point, the formation time of H⁺(H₂O)₄ becomes long compared with its lifetime with respect to thermal decomposition and its relative concentration decreases also. Implications of these results with respect to studies of the D-region are discussed.